Re: how to list all of the real numbers

On Aug 27, 12:03 am, Virgil <vir...@xxxxxxxxxxx> wrote:
In article <1188193676.421234.264...@xxxxxxxxxxxxxxxxxxxxxxxxxxxx>,
"Ross A. Finlayson" <r...@xxxxxxxxxxxxxxx> wrote:

Are you asking me? Others here would probably be familiar with the
sketch of the argument that any theory containing all elements that ZF
contains would find that the domain of discourse of ZF, its
"universe", is the Russell set and thus contains itself, so, there is
a universe in ZF, its own. (ZF is inconsistent.)

ZF may have a "universe", but there is no way in ZF to establish that
that universe is a set.

NBG has a universe wish is specifically not a set.



Pedagogically it is acknowledged that the universe is not deemed a set
in regular set theories (those with the axiom of regularity). While
that may be so, the set-theoretical universe is generally introduced
for fundamental considerations of set theory, "naive" set theory.

So don't be naif.

That is to say, in classrooms around the world the universe and sets'
complements within the universe (domain of discourse) are considered
as sets, casually and usually without complication. (If you find that
offensive please do.)

Until students newly being introduced to sets have some idea about what
sets can do, it is pedagogically foolish to rub their noses in what sets
do not do.

And naive set theories in which everything is constrained to some
relatively small set as a universe, as in, say, Euclidean plane
geometry, is quite valid, both pedagogically and mathematically.

Others here would probably be familiar with the sketch of the argument
that any theory containing all elements that ZF contains would find
that the domain of discourse of ZF, its "universe", is the Russell set
and thus contains itself, so, there is a universe in ZF, its own. (ZF
is inconsistent.)

(In the dually-self-intraconsistent null axiom theory the universal
set is a set, the ur-element, void's complement. Set theories where
it's not a set are not pure set theories.)

So, considering some fundamental physical particles as basically point
particles, the functions between them represent the interactions
between them, as do those between theirs and etcetera ad infinitum,
so, the physical universe contains infinitely many of those types of
things, and, the functions between them are again as well part of the
same thing, thus, the physical universe is an example of self-same
set: and powerset. (It is not actually novel to suggest the physical
universe contains itself, in a sense.)

Returning to the subject of "how to list all of the real numbers", I
encourage you to consider further the natural/unit equivalency
function, in regards to the discussion as above. Is it a function?
(Axiomatic support is readily supplied.) As a function, is it a
cumulative distribution function, besides being dense in the unit
interval and having a range that is naturally well-ordered, by the
natural integers in the real numbers natural total linear ordering?

It's quite a simple thing.

Ross

--
Finlayson Consulting

.

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